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A058711
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Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n).
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7
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1, 1, 1, 1, 4, 1, 1, 14, 11, 1, 1, 51, 106, 26, 1, 1, 202, 1232, 642, 57, 1, 1, 876, 22172, 28367, 3592, 120, 1, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1
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OFFSET
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1,5
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COMMENTS
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The old references had some typos, some of which were corrected in the recent ones. Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51; T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.
This triangular array is the same as A058710 except that it has no row n = 0 and no column k = 0.
(End)
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LINKS
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FORMULA
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T(n,1) = 1 for n >= 1.
T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.
T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]
(End)
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EXAMPLE
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Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
1;
1, 1;
1, 4, 1;
1, 14, 11, 1;
1, 51, 106, 26, 1;
1, 202, 1232, 642, 57, 1;
1, 876, 22172, 28367, 3592, 120, 1;
1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
...
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CROSSREFS
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Same as A058710 (except for row n=0 and column k=0).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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